Theorem arwhoma 17945

Description: An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
arwrcl.a	⊢ 𝐴 = (Arrow‘𝐶)
arwhoma.h	⊢ 𝐻 = (Homa‘𝐶)

Assertion

Ref	Expression
arwhoma	⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹)))

Proof of Theorem arwhoma

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		arwrcl.a	. . . . . . 7 ⊢ 𝐴 = (Arrow‘𝐶)
2		arwhoma.h	. . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶)
3	1, 2	arwval 17943	. . . . . 6 ⊢ 𝐴 = ∪ ran 𝐻
4	3	eleq2i 2824	. . . . 5 ⊢ (𝐹 ∈ 𝐴 ↔ 𝐹 ∈ ∪ ran 𝐻)
5	4	biimpi 215	. . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ∪ ran 𝐻)
6		eqid 2731	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
7	1	arwrcl 17944	. . . . . 6 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat)
8	2, 6, 7	homaf 17930	. . . . 5 ⊢ (𝐹 ∈ 𝐴 → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
9		ffn 6673	. . . . 5 ⊢ (𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V) → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)))
10		fnunirn 7206	. . . . 5 ⊢ (𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)) → (𝐹 ∈ ∪ ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧)))
11	8, 9, 10	3syl 18	. . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∈ ∪ ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧)))
12	5, 11	mpbid 231	. . 3 ⊢ (𝐹 ∈ 𝐴 → ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧))
13		fveq2 6847	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
14		df-ov 7365	. . . . . 6 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
15	13, 14	eqtr4di 2789	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
16	15	eleq2d 2818	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹 ∈ (𝐻‘𝑧) ↔ 𝐹 ∈ (𝑥𝐻𝑦)))
17	16	rexxp 5803	. . 3 ⊢ (∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧) ↔ ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦))
18	12, 17	sylib 217	. 2 ⊢ (𝐹 ∈ 𝐴 → ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦))
19		id 22	. . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ (𝑥𝐻𝑦))
20	2	homadm 17940	. . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (doma‘𝐹) = 𝑥)
21	2	homacd 17941	. . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (coda‘𝐹) = 𝑦)
22	20, 21	oveq12d 7380	. . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → ((doma‘𝐹)𝐻(coda‘𝐹)) = (𝑥𝐻𝑦))
23	19, 22	eleqtrrd 2835	. . . 4 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹)))
24	23	rexlimivw 3144	. . 3 ⊢ (∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹)))
25	24	rexlimivw 3144	. 2 ⊢ (∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹)))
26	18, 25	syl 17	1 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  ∃wrex 3069  Vcvv 3446  𝒫 cpw 4565  ⟨cop 4597  ∪ cuni 4870   × cxp 5636  ran crn 5639   Fn wfn 6496  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362  Basecbs 17094  dom_acdoma 17920  cod_accoda 17921  Arrowcarw 17922  Hom_achoma 17923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-1st 7926  df-2nd 7927  df-doma 17924  df-coda 17925  df-homa 17926  df-arw 17927

This theorem is referenced by:  arwdm  17947  arwcd  17948  arwhom  17951  arwdmcd  17952  coapm  17971